Problem 157 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(A_{n}\) be the area bounded by the curve \(y=(\tan x)^{n}\) and the lines \(x=0, y=0\) and \(x=\frac{\pi}{4}\), then for \(n>2\), (A) \(A_{n}+A_{n-2}=\frac{1}{n-1}\) (B) \(A_{n}+A_{n+2}=\frac{1}{n+1}\) (C) \(\frac{1}{2 n+1}

Step-by-Step Solution

Answer

(A) \(A_{n} + A_{n-2} = \frac{1}{n-1}\)

Step 1: Understand the Integral Expression for Area

The area bounded by the curve, x-axis, and the lines is given by the integral:\[A_n = \int_{0}^{\frac{\pi}{4}} (\tan x)^n \, dx\] where \( 0 \leq x \leq \frac{\pi}{4} \).

Step 2: Use the Reduction Formula for Evaluation

For an integrand of the form \((\tan x)^n\), use the reduction formula for integration:\[I_n = \int (\tan x)^n \, dx = \frac{(\tan x)^{n-1}}{n-1} - \int (\tan x)^{n-2} \, dx\]This formula allows us to express \(A_n\) in terms of \(A_{n-2}\).

Step 3: Integrate Using Limits

Let's assume:\[A_n = \int_{0}^{\frac{\pi}{4}} (\tan x)^n \, dx = \frac{(\tan x)^{n-1}}{n-1} \bigg|_0^{\frac{\pi}{4}} - \int_{0}^{\frac{\pi}{4}} (\tan x)^{n-2} \, dx\]Since \(\tan\left(\frac{\pi}{4}\right) = 1\), evaluating this gives:\[A_n = \frac{1}{n-1} - A_{n-2}\]

Step 4: Solve Given Conditions

Manipulate the equation \(A_n = \frac{1}{n-1} - A_{n-2}\) to find possible expressions:- Multiply throughout by \((n-1)\) and solve: \((n-1)A_n + (n-1)A_{n-2} = 1\)- Hence, \(A_n + A_{n-2} = \frac{1}{n-1}\)This matches option (A).

Step 5: Confirm the Correct Choice

Since the relation \(A_n + A_{n-2} = \frac{1}{n-1}\) fits exactly with option (A), options (B), (C), and (D) do not satisfy the condition derived from this reduction solution.So, the correct answer is (A).

Key Concepts

Understanding Definite Integral

In the realm of integral calculus, a definite integral is a fundamental concept that helps us compute the total accumulation of quantities, such as area under a curve within a specific interval. The definite integral is denoted by:

• The integral sign (∫),
• The function or formula being integrated (the integrand), which describes the curve,
• The limits of integration, which specify the interval \([a, b]\) over which you are finding the area.

In our exercise, the goal is to find the area bounded by the curve of \( y = (\tan x)^n \), the x-axis, and between the limits of \( x = 0 \) and \( x = \frac{\pi}{4} \). Essentially, the definite integral in this context is calculating the total area enclosed under the specified curve and lines within the given range. This helps us understand how the function behaves over that interval and allows us to summarize the behavior through the calculated area value.
The mathematical form representing this area is:\[A_n = \int_{0}^{\frac{\pi}{4}} (\tan x)^n \, dx\]For students, the definite integral not only represents a geometric visualization of area but also a key tool to transpose a curve into numerical form understandable within the given limits.

Applying the Reduction Formula

The reduction formula is an essential tool in calculus, providing a systematic technique to evaluate complex integrals like the one in our exercise. When dealing with an integrand of form \( (\tan x)^n \), the reduction formula helps simplify the integral by expressing it in terms of a simpler or already known integral. This recursive approach breaks down the problem gradually until it becomes straightforward to compute.
The formula for the reduction of \( \int (\tan x)^n \, dx \) is:\[I_n = \int (\tan x)^n \, dx = \frac{(\tan x)^{n-1}}{n-1} - \int (\tan x)^{n-2} \, dx\]This equation allows for integrating \( (\tan x)^n \) by reducing its power and in turn simplifying the calculation process. By recurring it, you translate the original problem into an evaluable and more manageable form. In our case, it directly leads us to the relationship \( A_n = \frac{1}{n-1} - A_{n-2} \), bridging the current integration task to a previously evaluated one. Understanding and applying this formula is crucial step-by-step to progressing in calculus.

Exploring Bounded Area Under Curve

The concept of the bounded area under a curve refers to the region enclosed alongside a curve, typically between the curve itself and the x-axis, and in our context, between vertical lines at \( x = 0 \) and \( x = \frac{\pi}{4} \). Calculating this area gives insight into the quantitative accumulated 'weight' of the function over the specified range.

• The area is effectively computed using definite integrals, as explained, specifically using the integral \( \int_{0}^{\frac{\pi}{4}} (\tan x)^n \, dx \).
• The result signifies the measure of the section of the plane that this function influences or covers within the specified bounds.
• It is essential as it extends practical utility in many applied fields where understanding the magnitude or measure of coverage is necessary.

In our specific problem, the bounded area provides a critical constraint and insight into the behavior of \( \tan(x)^n \) for the specified n-values and integrates mathematical theory with tangible outcomes. Recognizing these conceptual interpretations and computations heightens the full appreciation of integral calculus and its real-world applications.